On the death of his successor Charles Eugene Delaunay (1816-1872), he was reinstated by Thiers, but with authority restricted by the supervision of a council.
Delaunay, developed in his Theorie du mouvement de la lune (2 vols., 1860, 1867), because it contains a germ which may yet develop into the great desideratum of a general method in celestial mechanics.
It was found on comparing the results of Hansen and Delaunay that there are some outstanding discrepancies which are of sufficient magnitude to demand the attention of those interested in the mathematical theory of the subject.
This diminution was soon fully confirmed by others, especially Delaunay, although for some time Pontecoulant stoutly maintained the correctness of the older result.
In 1856 William Ferrel showed that the action of the moon on the ocean tidal waves would result in a retardation of the earth's rotation, a result, at first unnoticed, which was independently reached a few years later by Delaunay.